I feel your pain. The same thing happened to me when I had thermal reconditioning done. My hair was falling out in clumps. I mean, my bathroom floor looked like a rug whenever I would brush my hair. Now, are you sure the hair is falling out or just breaking off? If you can see a white bulb on the hair strand, the is falling out. If not then is breaking off. If you can see the bulb then damage has been caused to the actual hair bulb and I doubt even rogaine will help. Another thing even if rogaine was to help, you have to keep using it forever, because as soon as you stop the hair you have managed to grow will fall out again.
Now, if it is just breakage, protein treatments and deep conditioning will help keep the hair you still have. Redken Deep Fuel, Redken Cat, Nexxuss Emergencee, or Joico Deep reconstructor are all great. Start doing the protein treatments once a week for the first 4 weeks and then once a month after that. Also deep condition after the protein treatments. Try Redken All soft heavy cream. Put it on your hair, cover it with a shower cap and apply heat for 15 minutes. If you don't deep condition after the protein treatment your hair will feel like straw.
There is nothing you can do about the hair that has broken off, except wait for it to grow back. I know is horrible. It took me two years to grow my hair back. I had patches of broken hair and a bald spot on my frontal hairline. I couldn't wear my hair down for about a year, because the hair was visible uneven. So I had to wait for it to be long enough to add layers and blend it in.
That happens when the stylist performing the service is not an expert. The process was done incorrectly. I never had it done again. I now use Rusk Radical Anticurl. It is thio based just like the products used to perform TR. However is more gentle. No heat is used. The curl is texturized not permanently straighten like with TR. Once I blow dry and/or flat iron, my hair stays super straight, smooth and frizz free no matter what. It is healthy and beautifull.
Don't stress over, it will just make matters worse. Start doing the treatments and slowly cutting off the damaged hair. Your hair will get back to normal.
Good luck and great hair
Shary